Worksheet: Equation of a Plane: Intercept and Parametric Forms

In this worksheet, we will practice finding the equation of a plane in different forms, such as intercept and parametric forms.

Q1:

Find the general equation of the plane 𝑥=4+7𝑡+4𝑡, 𝑦=−3−4𝑡, 𝑧=1+3𝑡.

A𝑥+4𝑦+7𝑧+16=0

B𝑥+12𝑦−28𝑧=0

C3𝑥+3𝑦−7𝑧+4=0

D12𝑥+4𝑦+7𝑧−43=0

E𝑥−12𝑦+28𝑧−16=0

Q2:

Write, in intercept form, the equation of the plane 16𝑥+2𝑦+8𝑧−16=0.

A𝑥1+𝑦8+𝑧2=16

B𝑥16+𝑦2+𝑧8=1

C𝑥1+𝑦8+𝑧2=1

D𝑥16+𝑦16+𝑧16=1

E𝑥16+𝑦2+𝑧8=16

Q3:

What is the length of the segment of the 𝑦-axis cut
off by the plane 5𝑥−4𝑦−3𝑧+32=0?

A18 of a length unit

B8 length units

C32
length units

D4 length units

Q4:

Find the general form of the equation of the plane which intersects the coordinate axes at the points (2,0,0), (0,8,0), and (0,0,4).

A4𝑥+𝑦+2𝑧+8=0

B𝑥+4𝑦+2𝑧+7=0

C4𝑥+𝑦+2𝑧−8=0

D𝑥+4𝑦+2𝑧=0

Q5:

Given that the plane 2𝑥+6𝑦+2𝑧=18 intersects the coordinate axes 𝑥, 𝑦, and 𝑧
at the points 𝐴,
𝐵, and 𝐶, respectively, find the area of △𝐴𝐵𝐶.

A27√11

B2√19

C27√112

D3√192

E3√152

Q6:

Determine the general equation of the plane that intersects the negative 𝑥-axis at
a distance of 2 from the origin, intersects the positive 𝑧-axis at a distance of 3 from the origin,
and passes through the point 𝐶(9,−4,−4).

A2𝑥+3𝑧−6=0

B12𝑥+41𝑦−8𝑧+24=0

C11𝑥−4𝑦−4𝑧+12=0

D12𝑥−41𝑦−8𝑧−24=0

E9𝑥−4𝑦−7𝑧+18=0

Q7:

Find the general equation of the plane that passes through the point (8,−9,−9) and cuts off equal intercepts on the three coordinate axes.

A𝑥+𝑦+𝑧+10=0

B𝑥+𝑦+𝑧−648=0

C8𝑥−9𝑦−9𝑧=0

D𝑥+𝑦+𝑧−10=0

E8𝑥+𝑦+𝑧=0

Q8:

Find the equation of the plane cutting the coordinate axes at 𝐴, 𝐵, and 𝐶,
given that the intersection point of the medians of △𝐴𝐵𝐶 is (𝑙,𝑚,𝑛).

A𝑙𝑥+𝑚𝑦+𝑛𝑧=1

B𝑥+𝑦+𝑧=𝑙+𝑚+𝑛

C𝑙𝑥+𝑚𝑦+𝑛𝑧=3

D𝑥𝑙+𝑦𝑚+𝑧𝑛=3

E𝑥𝑙+𝑦𝑚+𝑧𝑛=1

Q9:

Find the equation of the plane whose 𝑥-, 𝑦-, and 𝑧-intercepts are −7, 3, and −4, respectively.

A−𝑥4−𝑦7−𝑧4=1

B𝑥3−𝑦7−𝑧4=1

C−𝑥7−𝑦4+𝑧3=1

D−𝑥7+𝑦3−𝑧4=1

Q10:

What is the length of the segment of the 𝑥-axis cut off by the plane 6𝑥+3𝑦+5𝑧=4?